Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Bonus-Malus Scales 197
under the financial balance constraint
Er lin
L = E = 1
⇔ E = + EL ⇔ = E − EL
It suffices to minimize
[ ( (
))]
˜ = E exp − c − E − L − EL
Differentiating ˜ with respect to and equating to zero yields
[
(
E L − EL exp − c ( − E − L − EL ))] = 0
⇔
∫ +
0
s∑
( (
))
l − EL exp − c − E − l − EL l dF = 0
l=0
which has to be solved numerically to get the value of (and hence of ). Convenient
starting values for the numerical search are provided by (4.17).
4.6.4 Numerical Illustration
In this section, we give numerical examples of computation of bonus-malus scales
when using an exponential loss function. We compare the results with those obtained
previously.
In order to be able to compare the results, we have computed the relativities associated
with the same severity factor c = 1. These results are given in Table 4.13 for the −1/top,
−1/ + 2 and −1/ + 3 systems and Portfolio A. Specifically, Table 4.13 gives the relativities
obtained with an exponential loss function with severity parameter c = 1, with and without a
priori risk classification, as well as the analogues for the −1/ + 2 and −1/ + 3 systems. As
was the case with the quadratic loss function, we see that a priori risk classification reduces
the dispersion of the relativities.
We observe that the relativities computed when no a priori ratemaking is in force give
bigger bonuses when the policyholders are in level 0 but also impose bigger maluses in the
other levels. Indeed, when no a priori ratemaking is in force, the a posteriori correction
must be more severe in order to distinguish between good and bad drivers. On the contrary,
when an a priori ratemaking is in force, the correction applied with the a posteriori tariff
must be softer because a greater part of the risk is already taken into account in the a priori
ratemaking.
Influence of the Loss Function
We can also compare the results obtained when using the quadratic loss function and
when using the exponential loss function (with different values of c). These are given in